A Reconstruction of the Differential Analyzer in Meccano

FEATURE A Reconstruction of the Differential Analyzer in Meccano

The challenges of torque amplifiers, integrators, and backlash

By Tim Robinson

was first introduced to Meccano, a child’s educational construction set cre- ated in the United Kingdom, at about the age of six and quickly became fascinated with it as a medium for constructing working mechanisms and small machines. Over the next ten years or so, I gath- Iered quite a large collection. I first attempted to construct a differential analyzer in Meccano around 1971. I had just encountered calculus in high school and at the same time I started to develop an interest in computers. One of the first books I read on computers included a chapter on analog computation, and on the differential analyzer in particular. Significantly, the book briefly men-

©DIGITALVISION tioned that simple differential analyzers had been constructed in Meccano in the 1930s. So began an interest that has remained with me for more than 30 years.

Early Attempts My early attempts were not very successful because of the difficulty of constructing functioning torque amplifiers. When Hartree and Porter built the first Meccano differential analyzer at Manchester Universi- ty, their goal was to build a working machine quickly. They used Meccano simply because it was readily available and allowed them to avoid designing and custom machining most of the required parts. However, Hartree and Porter felt no constraint to stay within the limits of the Meccano system. In particular, they did not believe that adequate torque amplifiers could be made without custom machining. In contrast, as a Meccano enthusiast approaching the same subject, finding a solution within the system to what at first seems an insurmountable problem was part of the challenge.

0272-1708/05/$20.00©2005IEEE 84 IEEE Control Systems Magazine June 2005 half of which are nuts, bolts, and washers. Construction started about four years ago, after a flash of insight led me to a robust solution to the amplifier problem. Instead of arranging the input and output shafts of the amplifier to be colinear with one drum on each as Bush had done, I arranged the input shaft to run parallel to the output, with both drums running on the output shaft (Figure 2). The input is coupled to the input arms of the amplifier by means of a pair of gears, one fixed on the input shaft and the other free to rotate on the output shaft between the drums. Since the output shaft has the benefit of the amplifi- cation of torque compared to the input, any residual friction from the drums running directly on the output shaft is unimportant. While this feature was the key to constructing a working amplifier within the constraints of the Meccano system, another critical element turned out to be the Figure 1. Aerial view of the machine. Down the center is the choice of material for the friction bands. Particularly in the interconnect section in which shafts and gearing are placed second stage, the bands are under heavy load and can heat to “program” the machine for a particular equation. On the up significantly from the friction. If the band material is not left are the input table and dual output table, and on the right stable, the critical adjustment of tension in the bands will are four integrators. Each of the functional units can be be upset during operation. If the material stretches, exces- detached from the central interconnect for maintenance and transportation. (Photo: Michael Baxter.) sive backlash is introduced and the bands are likely to slip off the drums. If the material shrinks, then the bands tight- en, increasing friction further in a cycle of positive feedback Having failed in that first attempt, I revisited the differen- that leads to the amplifier seizing up. After experimenting tial analyzer periodically. The main obstacle to making the with many materials, I finally selected Dacron cord, a mate- amplifier is the lack of any form of concentric shafting sys- rial favored by kite flyers because of its light weight, great tem. In Bush’s prototype machine, the amplifiers use drums strength, and resistance to stretching. Dacron performs that rotate on hollow shafts rigidly fixed to the framework. extremely well for my torque amplifier design. This arrangement allows the input and output shafts to pass With these fundamental issues addressed, I evaluated a through with no friction coupling directly from the drums to number of design variations, varying in minor details most- the input. Copying the general layout in Meccano, but without ly to improve strength and rigidity. I steadily improved the the fixed sleeve between the input shaft and performance and finally arrived at a two- the drum, inevitably means that an unac- stage design that runs reliably for days at a ceptable torque is coupled from one of the time (Figure 3). The rest of the machine is drums directly to the input shaft. This cou- relatively straightforward by comparison. pling causes the integrator wheel to steadily slip in the direction of the drum’s rotation. Integrators By adding a counter-rotating element to the Integrators are the principal functional input shaft to at least partly compensate for units of a differential analyzer, and a few the undesirable friction, I succeeded in mak- details of the integrators in the model are ing a simple two-integrator demonstration worth highlighting (Figure 4). The output model. Although the amplifiers had only a of an integrator is taken from the integrat- single stage, with the use of ground glass ing wheel, which rests on the disk under instead of plain glass discs to increase the very light pressure. The available torque friction available at the integrator wheel, the is further reduced by any vibration when mechanism could be made to solve the sec- the machine is in operation. To ensure Torque amplifier ond-order equation for simple harmonic Figure 2. there is no slipping, friction on the shaft stage. The output shaft to the motion at least qualitatively. that couples the integrating wheel to the left carries both rotating drums. The input shaft runs input of the torque amplifier must be mini- A Successful Torque parallel on the right, carried in mal. Friction reduction is accomplished Amplifier rotating bearings, and couples by mounting the shaft in rotating bearings, The machine I have today (Figure 1) con- to the input arms through a driven from the output of the torque tains about 16,000 individual parts, about pair of gears. amplifier. Because the bearings rotate at

June 2005 IEEE Control Systems Magazine 85 the same rate as the shaft, friction is essentially reduced to Backlash zero. If the integrator wheel is lifted clear of the integrator In all mechanical systems, there is unavoidable backlash. disk and set in motion, the wheel will spin almost indefi- This fact is particularly true in the case of Meccano, which, nitely. Although the rotating bearings introduce the possi- being essentially a toy, is not manufactured to especially bility of feedback-path coupling from the output of the close tolerances. Starting from an integrating wheel, there amplifier back to the input, there appears to be no evi- is backlash in the torque amplifier as well as in the gears in dence of instability in practice. the shafts connecting the output of the amplifier to either a The integrator disks are made from window glass cut to lead screw or the disk of another integrator. In an analog a circle with a diameter of 11 in and attached to a standard machine, where variables are represented by shaft angles, Meccano component using hot glue. Glass is a convenient, backlash represents a significant source of error. In his readily available material that provides a hard-wearing, prototype, Bush introduced something he called a “front- flat surface. A large diameter provides greater accuracy, and lash” unit to compensate the backlash to first order. These this size was chosen because the longest Meccano units took the form of epicyclic gear assemblies, which screwed rod, used for the carriage lead screw, is 11-1/2 in. could be inserted as required in the interconnect. Whenever The disk is fairly heavy and must rotate very freely. To the direction of motion of the input to the frontlash unit facilitate free rotation, the builders of the early Meccano reverses, a small amount of extra motion is added to the differential analyzers again reached outside the system output to compensate for the motion lost to the backlash. and used commercial ball bearing assemblies to carry the The frontlash unit proved to be another difficult item to disk. I adopted an effective alternative solution, namely, model in Meccano, not because of any problem in princi- supporting the weight of the disk on three small wheels ple but because of the difficulty of creating a unit that can positioned near the periphery. The driving shaft provides operate smoothly under the heavy loading of actual opera- centering and the driving torque, but does not need to sup- tion. After a number of unsatisfactory attempts, I solved port the weight. The disk can be lifted off for cleaning. this problem by using a differential-based design, located The integrator carriage must be accurately positioned to between the two stages of each torque amplifier. By posi- represent the value of the function being integrated. Any play tioning the differential there, the frontlash unit operates in the lead screw directly affects the accuracy of the machine. under only a light loading since it is followed by the gain of To avoid play, two nuts are used on the screw, forced apart the second amplifier stage. The amount of compensation is by a compression spring. The force of the spring ensures that adjustable and can be set for a particular setup by measur- the defining nut always contacts the same side of the lead- ing the backlash empirically. screw threads. This device was used by Bush in the prototype and he referred to it as a “lashlock.” To eliminate end Input/Output Tables float in the bearing, a second spring preloads the bearing of In Bush’s prototype, the input and output tables each had the shaft that drives the screw. a fixed bed over which the cross hair or recording pen could be driven in two dimensions by a pair of lead screws. In Meccano, it is much easier to arrange for the cross hair or pen to move along a fixed axis and have the bed of the table move bodily in the perpendicular direction beneath it. This arrangement was used by Hartree and

Figure 3. Two-stage amplifier. This amplifier has two simi- lar stages connected in series for an overall gain of approxi- mately 10,000. A “frontlash” unit in the form of a differential gear can be seen at the upper left, in the path between the Figure 4. A pair of integrators, each incorporating a two- two stages. The amount of backlash to be compensated is set stage torque amplifier. Digital counters on the carriage lead by using the small hand crank. screws facilitate accurate setting of the initial conditions.

86 IEEE Control Systems Magazine June 2005 Meccano Differential Analyzer at VCF 7.0

n 6–7 November 2004, the Vintage Computer Festi- reliably for both days of the event. Although the chosen setup val held its seventh annual conference and exhibi- used only three of the available four integrators, the setup Otion, hosted by the Computer History Museum in was simple enough that visitors with an engineering or scien- Mountain View, California. This tific background could quickly year’s event attracted some 15 relate to it. An accompanying dis- speakers, 40 exhibitors, and 500 play board covered the history of attendees. The festival brings togeth- the prototype, reviewed the princi- er enthusiasts, historians, and inter- ples of operation, and documented ested members of the general the demonstration setup. Original public. The event has a broad focus, papers and articles demonstrating covering every aspect of computer comprehensive research completed hardware and software, providing the exhibit. the subject is at least 10 years old. A panel of judges assessed each Multiple conference tracks cov- exhibit according to a set of formal ered topics as diverse as “The IBM criteria. Awards were given for first, Two young boys are enthralled. Although too 360 Evolution and Revolution” and young to understand what the machine does, second, and third place in each “Using Vintage Computers in Com- the children are captivated by the vast number class. In addition, a number of spe- puter Forensics.” Exhibits ranged of moving parts. cial awards (independent of class) from a display of 60 years of palm- were offered. The Best in Show sized computational devices, award was taken by an outstanding through a broad assortment of exhibit of products from The Digital early and long-forgotten personal Group covering the period computers and workstations, to a 1974–1978. The differential analyz- recreation of a 1972 DEC-based er received 1st place in its class and data center, complete with multi- claimed special awards for “Best ple mag-tape drives, line-printers, Technology: Analog” and “Best and an array of interactive termi- Technology: Nonelectronic.” nals, running an equally ancient The differential analyzer exhibit version of UNIX. proved popular. Experts and the gen- Most of the exhibits were eral public, especially children, were Supporting display board. The display includes entered for judging in one of six equally fascinated. The combination pictures of Bush’s prototype, Hartree and Porter’s classes. To be eligible for judging, first Meccano machine, principles of operation, a of visible mechanical operation and exhibits must be fully operational, setup diagram for the equation being run for the medium of its construction must be running period software as demonstration, and a brief bibliography. immediately drew people toward the appropriate, and must operate with- machine. Although the machine is out error at the appointed time inherently slow, most people were before the judges. The author willing to take the time to watch the entered his Meccano model of solution steadily unfold, and many Bush’s 1930 differential analyzer in requested detailed explanations of the class “Re-creation, Emulation, or aspects of its operation. The mechan- Contemporary Enhancement.” The ical torque amplifiers received par- displayed machine had four integra- ticularly close attention. tors, input table, and dual-output At the conclusion of the show, table and, for demonstration pur- an additional prize, the “People’s poses, was configured to solve a Choice Award” was decided by The author at the input table while visitors look forced, damped simple harmonic on. The exhibit drew numerous onlookers who popular vote of the exhibition visi- oscillator equation. The Meccano were fascinated by the operation of the tors. The differential analyzer stole model operated continuously and machine. this one handsomely!

June 2005 IEEE Control Systems Magazine 87 Virtual Differential Analyzer Reconstruction

Original wheel-and-disc Animated wheel-and- integrator from Bush's dif- disc integrator. The ani- ferential analyzer on dis- mated integrator play at the MIT Museum. demonstrates the oper- This view shows the glass ation of this clever disc and the metal wheel mechanism, providing used to implement inte- insight into the opera- gration in the computer. tion of the differential analyzer.

Original Nieman torque Animated torque amplifier on display at the amplifier. By watching MIT Museum. This torque a short animation of the amplifier, based on con- torque amplifier in tra-rotating capstans, operation, students can amplifies the output from observe and learn the the integrator wheel and principles behind the drives the output gears of amplifier's operation. the integrator box.

echanical differential analyzers are often praised spline curves and meshes of polygons. The scene is illumi- for their educational value. Vannevar Bush tells nated with three sources of light, one directional source the story of a draftsman who learned differential and two ambient sources, to enhance the appearance of M the components. Texture maps of metal, wood, brick, and equations in mechanical terms from working on the construction and maintenance of the MIT differential analyzer floor tiles are used to add detail to the rendered objects. (see [1]). After the last differential analyzer at MIT was The complete model will be animated to show the oper- decommissioned, Warren Weaver wrote to Samuel Caldwell: ation of the machine solving differential equations. Several days of computer time are needed to produce the individual [I]t seems rather a pity not to have around such a frames for a single animation. The final video is assembled place as MIT a really impressive Analogue comput- using Adobe AfterEffects and compressed using the DivX er; for there is vividness and directness of meaning codec. Preliminary images and videos from this project are of the electrical and mechanical processes available on the Web [5]. involved... which can hardly fail, I would think, to have a very considerable educational value. A Digi- References tal Electronic computer is bound to be a somewhat [1] T. Robinson, “The Meccano set computers,” IEEE Contr. Syst. . abstract affair, in which the actual computational Mag., vol. 25, no. 3, pp. 74–83, June 2005 processes are fairly deeply submerged. [2] [2] L. Owens, “Vannevar Bush and the differential analyzer: the text and context of an early computer,” Technol. Culture, vol. 27, no. 1, pp. 63–95, Jan. 1986. In the place of an expensive reconstruction of the [3] D.A. Mindell, “MIT differential analyzer” [Online]. Available: original high-precision machinery [3], we have complet- http://web.mit.edu/mindell/www/analyzer.htm ed a virtual reconstruction of the Bush differential analyz- [4] V. Bush and H. Hazen, “The differential analyzer: A new machine er. Based on original descriptions and photographs of the for solving differential equations,” J. Franklin Inst., vol. 212, no. 4, pp. computer [4], a graphical model of the computer has 447–488, Oct. 1931. been constructed using Maya Complete 6.0, a three- [5] K.H. Lundberg and V. Pereverzev, “Vannevar Bush's differential analyzer” [Online]. Available: http://web.mit.edu/klund/www/analyzer/ dimensional (3-D) modeling software package. The analyzer is modeled with a physical description of —Kent Lundberg and Vitaliy Pereverzev

88 IEEE Control Systems Magazine June 2005 Porter in the first Meccano model, and adopted in my localized. Each segment can be removed independently design. I have, however, tilted the table beds to 45° to to permit easy changing of gears and interconnection make the operator’s task more pleasant. The tables are points, just as on Bush’s prototype. sized to accept standard 11 in × 17 in graph paper. Once an equation has been set up on the machine, initial conditions must be set prior to each run. To facilitate Overall Layout initialization, all lead screws can be disconnected from the On this incarnation of the model, I set out with the goal of main interconnect by friction clutches, thus permitting reproducing Bush’s machine configuration as closely as them to be set to the appropriate starting positions. Each possible. In particular, the design anticipates a total of six lead screw is coupled to a decimal rotation counter, which integrators, although so far I have completed only four. provides settings that are accurate to a fraction of a turn. The prototype used a monolithic construction literally A small motor drive is provided at each integrator lead bolted to a concrete floor. Similarly, when Hartree and screw to reduce the amount of hand cranking needed to Porter built the first Meccano model, they were not con- set initial conditions. All lead screws include limit switches, cerned with portability, and used a large sheet of ply- which cut power to the independent variable motor in the wood as a base to which the various Meccano parts could event that any unit exceeds its mechanical limits. Without be firmly screwed down. I decided that I wanted a limit switches, excessive rotation can induce a severe machine that could easily be transported and reconfig- strain on the torque amplifier driving the lead screw that ured for demonstration purposes (see “Meccano Differen- hits the limit, usually resulting in damage. tial Analyzer at VCF 7.0.”), so I constructed some lightweight custom tables and arranged the framework of Future Plans the machine to be modular, resting on the tables on I plan a number of enhancements to my design. Foremost vibration-absorbing rubber feet. Although this approach is the addition of more integrators. Realistic problems requires far more Meccano parts, modularity is much require more integrators than the simple mathematical more in the true spirit of the medium. It also makes main- order of the equation because additional integrators are tenance easier, since individual units can be readily used as function generators and sometimes as multipliers. removed to a workbench. Indeed, the central table was initially sized assuming an I wanted to create more than a simple demonstration eventual configuration with six integrators. Next in line is a model. Specifically, my goal was to build a machine to sim- second input table configured to allow optional use as a plify the work required to set up a new equation and per- multiplier. With these additions, the capacity of the form repeated runs with varying initial conditions; that is, machine will be quite comparable to Bush's prototype and a machine that has good “usability.” In this respect, the to other reconstructions (see “Virtual Differential Analyzer 1930s Meccano models fell somewhat short of the full-scale Reconstruction”). prototype, sometimes requiring modifications on the fly as new equations were being set up. Following Bush’s layout, my machine has a central interconnect section (which can Tim Robinson ([email protected]) retired in 2003 from be split into subsections for transport), into which the Broadcom Corporation, where, as senior director of engi- integrators and input/output tables can be plugged. One neering, he was responsible for the development of Broad- improvement that subsequent full-scale machines adopted com’s range of WiFi (802.11a/b/g) wireless networking relative to Bush’s prototype was the provision of a second chipsets. He holds bachelor’s and master’s degrees in level of bus shafts, which adds considerable flexibility in physics from Oxford University. He entered the computing setting up a particular equation. I decided to adopt a simi- field in 1980 in the United Kingdom, where, as cofounder of lar scheme. High Level Hardware Ltd., he designed a user-micropro- Scaling up the machine from a simple two-integrator grammable computer system for developing novel pro- demonstration model introduces new challenges. The gramming languages. In 1989, he moved to the San much greater amount of interconnect required for a real- Francisco Bay Area, where he has held senior engineering istic configuration increases the amount of friction and, positions at a number of Silicon Valley startup companies. with it, the loading on the torque amplifiers. Some bus He maintains a strong interest in the early history of com- shafts often need to run the entire length of the machine, puting, particularly mechanical computing devices, and is which is over 10 ft for the six-integrator configuration. actively involved in the restoration of these early machines This scale calls for great care and patience in construc- and in the construction of working replicas of Charles Bab- tion to achieve sufficient rigidity and accurate alignment bage’s conceptual designs. Additional interests include of components. The bus shafts are in fact made up of music, Meccano, and current developments in physics and much shorter axles with couplings between the sections. cosmology. Tim can be contacted at 216 Blackstone Dr., This design is essential to allow “programming” to be Boulder Creek, CA 95006 USA.

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